Question

Is lR^* cyclic? Try to prove your answer. Hint: Suppose k is a generator of lR^*

Answer 1

wth is IR?

Answer 2

do you mean \(\mathbb{R}\)?

Answer 3

Yes

Answer 4

But I'm not sure what the * means.

Answer 5

It's A7 chapter 11

Answer 6

They have more information and a small picture.

Answer 7

If k<1, then k>k^2>k^3>...

Answer 8

If k>1, then k<k^2<k^3<...

Answer 9

suppose to the contrary \(\mathbb{R}\) is cyclic with generator \(k\), then \(\mathbb{R}=\{0,k,k^2,k^3,....\}\) but by density for any \(k^i\ne k^j\) there exists \(r\in \mathbb{R}\) s.t. \(k^i<r<k^j\). A contradiction.

Answer 10

hmm the fact that they are giving other stuff makes me think they dont want you to use density.

Answer 11

Let me look...

Answer 12

My wife needs me to fix something. Are you coming out tomorrow?

Answer 13

Is there another way to prove this without using density?

Answer 14

im sure

Answer 15

Yes.

Answer 16

But what is r star?

Answer 17

you can tell me the other ones in there, and ill know what we are doing when you get there.

Answer 18

just postive R

Answer 19

with what operation?

Answer 20

addition I think. Look in the book in the section where it defines groups. It tells you some basic notation they use.

Answer 21

My problems are A7, B4, C5, E4, and F4 Chapter 11. and on cosets.

Answer 22

sweet. ok wife yelling at me. sorry. ill be there at 9

Answer 23

Okay thanks Zach.

Answer 24

for sure